J. Chem. Sci. Vol. 128, No. 10, October 2016, pp. 1527–1536. c Indian Academy of Sciences. Special Issue on CHEMICAL BONDING DOI 10.1007/s12039-016-1172-3 PERSPECTIVE ARTICLE

Bader’s Theory of Atoms in Molecules (AIM) and its Applications to Chemical Bonding

P SHYAM VINOD KUMAR, V RAGHAVENDRA and V SUBRAMANIAN∗ Chemical Laboratory, CSIR-Central Leather Research Institute, Adyar, Chennai, Tamil Nadu 600 020, India e-mail: [email protected]; [email protected]

MS received 22 July 2016; revised 17 August 2016; accepted 19 August 2016

Abstract. In this perspective article, the basic theory and applications of the “Quantum Theory of Atoms in Molecules” have been presented with examples from different categories of weak and hydrogen bonded molecular systems.

Keywords. QTAIM; non-covalent interaction; chemical bonding; H-bonding; electron density

1. Introduction account of the bonding within a molecule or crystal. Specially, the derivative of electron density on BCP It is now possible to define the structure of molecules is zero. Bader’s group has made several seminal con- quantum mechanically with the help of Bader’s Quan- tributions to the development of the QTAIM theory tum Theory of Atoms in Molecules (QTAIM).1,2 This and its applications to unravel chemical bonding.10 13 theory has been widely applied to unravel atom-atom Popelier and coworkers have employed the QTAIM interactions in covalent and non-covalent interactions to address several issues in chemistry.14 16 Particu- in molecules,3 molecular clusters,4 small molecular larly, they have demonstrated the possibility of devel- crystals,5 proteins,6 DNA base pairing and stacking.7 oping structure-activity-relationship to predict various Basic motive of QTAIM is to exploit charge density or physico-chemical properties.17 They have used the the- electron density of molecules ρ(r; X) as a vehicle to ory of Quantum Chemical Topology (QCT), to provide study the nature of bonding in molecular systems. The ab initio descriptors that are able to accurately predict electron density (ED) in general can be defined as,2,8,9 pKa values for 228 carboxylic acids.18,19 Hydrogen-   ∗ bond strengths have been described in terms of basici- ρ (r; X) = N dτ ψ (x; X)ψ(x; X) (1) ties with the help of several different scales. Bohorquez, Boyd, and Matta have illustrated the concept of deriving where N is the no. of electrons, x is the electronic coor- molecular model with quantum chemical bonding with dτ dinates, X is the nuclear coordinates and represents the help of the theory of QTAIM.1 Matta and cowork- the volume element of the system under consideration. ers have characterized the tri-hydrogen bond based on The ED distribution is predominantly affected by the topology of ED.20 interaction between two nuclei and hence the chem- Several groups have applied the QTAIM for unrav- ical bonding. It is found from previous reports that elling the non-covalent interactions such as weak van 4 5 both theoretical and experimental electron densities of der Waals, π...π,X-H...π, conventional hydrogen molecules can be used to gain insight into the nature of bonding, cation...π interactions, halogen bonds, etc. the chemical bonding. In fact, the topological analysis and also other applications in chemistry.3,21 24 All these of ED distributions has expanded beyond the territories interactions have been vividly characterized with the of theoretical chemistry to the X-ray crystallography. help of theory of QTAIM. It is not overstating that The topological properties of ED and its derivatives are QTAIM has completely provided new dimensions to the found to be immensely useful in delineating the con- concept of hydrogen bonding. Grabowski and cowork- cept of the bonding through bond paths and bond crit- ers have made several significant contributions to the 2,3 ical points (BCPs). The molecular graph and the ED characterization and understanding of hydrogen bond- along with its derivatives at the BCPs provide a precise ing interactions in various systems with the help of QTAIM.25,26 It is evident from previous studies that ∗For correspondence Bader’s topological parameters exhibited interrelation- Celebrating 100 years of Lewis Chemical Bond ship with geometrical parameters. For example, the 1527 1528 P Shyam Vinod Kumar et al. interrelationship between atom-atom distance and the is a linear relationship between interaction energy (IE) 3 ED at the BCP has been investigated. This relationship and ρ(rc) at the BCP for diverse class of molecular revealed that ED at the BCP provides a good measure systems.30 It is also evident from that report that weak, of strength of interaction between two molecular sys- moderate, strong and very strong hydrogen bonds can tems. It is also important to mention that QTAIM anal- be clearly described with the help of ED and its Lapla- ysis provided clear description of degree of covalency cian at BCPs. The relationship between Stabilization in the hydrogen bonding interaction. Espinosa and Energy (SE) and ED at BCP explain the smooth tran- coworkers27,28 have used parameters derived from the sition in the hydrogen strength from van der Waals to QTAIM analysis to elicit degree of covalency in hydro- covalent limit. gen bonding interaction. A measure of the strength The detailed theory of atoms in molecules has been 2a (com) of hydrogen bonding interaction has been devel- explained in the Bader’s Book. Other monographs oped with the help of QTAIM derived topological and on the QTAIM also provide basic theory and its geometrical parameters.29 It is given as: applications.31,32 In this perspective article, the basic theory and applications of QTAIM in eliciting the weak,  ={[(r −r0 )/r 0 ]2 +[(ρ0 −ρ )/ρ 0 ]2 van der Waals, hydrogen bonding and ionic bonding com A−H A−H A−H A−H A−H A−H interactions in various molecules and molecular clus- 2 2 0 2 0 2 1 ters have been described with suitable examples. Since +[(∇ ρ − −∇ ρ )/∇ ρ ] } 2 (2) A H A−H A−H voluminous information is available on this topic, the comprehensive coverage of theory and applications of r ρ ∇2ρ where A−H, A−H and A−H represent proton donat- QTAIM is beyond the scope of this perspective article. ing bond involved in H-bonding, the bond length, elec- However, important aspects of the theory are provided tronic density at A-H bond critical point and Laplacian in this article. r0 ρ0 ∇2ρ0 of this density respectively. A−H, A−H and A−H correspond to the same parameters of the A-H bond 1.1 The basic theory of quantum theory of atoms involved in H-bond formation or any other nonbonded in molecule interaction. This complex parameter clearly describes the cova- It is well known that the ED distribution of a molecular lency of hydrogen bonding interaction. Higher values of system is the physical manifestation of the forces act- 2 com indicate lengthening of the proton donating bond ing within the molecular system. The attractive force and shorter proton-acceptor distances. Higher value of exerted by the nuclei determines the most important com implies greater binding energy and negative value topological properties of ED of multi-electronic sys- of the total electron energy density at hydrogen bond tem. The ED exhibits local maxima only at the posi- critical point(HBCP) and hence the degree of cova- tions of the nuclei. Exceptions to this have also been lency of hydrogen bonding interaction. In addition to observed under certain circumstances. It is possible to QTAIM, electron delocalization function (ELF), Nat- recognize atomic forms within the molecules with the ural Bond Orbital analysis (NBO) and energy decom- help of local maxima exhibited by the ED at the nuclei. position analysis (EDA) are also useful in describing The ED distribution, ED relief map and molecular the covalency of hydrogen bonding interaction.2 In graph of benzene are shown in Figure 1, which clearly understanding degree of covalency of hydrogen bond- reveal the presence of local maxima at the position ing interaction, Parthasarthi et al., have shown that there of nucleus.

Figure 1. (a) Electron density contour map; (b) Relief map; (c) Molecular graph of Benzene. Atoms in Molecules and Chemical Bonding 1529

1.2 Topography of electron density distribution its associated basin.2 It can be seen from Figure 1c and critical points that (3, −1) CP is found between every pair of nuclei which are linked by a chemical bond in the benzene It can be inferred from Figure 1 that the ED exhibits molecule. It is found from detailed analysis of topolog- a maximum, a minimum, or a saddle point in space. ical features of ED that pairs of gradient paths which These special points are referred to as Critical Points originate at each (3, −1) CP and terminate at the neigh- ρ( ) (CPs). At this point, the first derivatives of rc van- boring attractors.2 These two unique paths describe a ∇ρ( ) = ∇ρ( ) ishes, i.e., rc 0, where rc is given in line through the charge distribution connecting neigh- equation (3) and rc is the CP. boring nuclei along which ρ is a maximum with respect ∂ρ ∂ρ ∂ρ to any neighboring line. Such a line is known as ∇ρ(r ) = i + j + k c ∂x ∂y ∂z (3) atomic interaction line in the topological analysis of ED distribution.2 The presence of a (3, −1) CP and It is well known that a maximum or a minimum or related atomic interaction line highlights that electronic an extremum is determined by the sign of its second charge density is accumulated between the nuclei that derivative at this point. Therefore, it is necessary to are bonded. The existence of an atomic interaction line explore the second order derivatives of ED. For an in an equilibrium geometry satisfies both necessary and arbitrary choice of coordinate axes, nine second order sufficient conditions that the atoms be bonded to one derivatives are possible. It is represented in the form of another. Hence, it is called as bond path and the (3, −1) a real and symmetric matrix known as the Hessian of CP is referred to as bond critical point (BCP). All these ρ(rc). It can be diagonalized with the help of unitary topological features define molecular graph. transformation to obtain eigenvalues; λ1, λ2 and λ3 are Other CPs of rank three arise due to the particular the principal axes of curvature because the magnitude geometrical arrangements of bond paths. In the case of of the three second derivatives of ρ(rc)calculated with benzene, the bond paths are connected to form a ring. respect to these axes are extremized. The (3, +1) CP found in the interior of the ring is known as ring critical point (RCP). A ring is defined as a part of ∂2ρ(r ) ∂2ρ(r ) ∂2ρ(r ) 2 ∇2ρ(r ) = c + c + c = λ +λ +λ a molecular graph, which bounds a ring surface. If the c dx2 dy2 dz2 1 2 3 bond paths enclose the interior of a molecule with ring (4) surfaces, then the Cage Critical Point (CCP) (3, +3) The CPs are designated as (ω, σ), where ω is rank of is observed.2 CP and σ is its signature. The rank of a CP is equal to the number of non-zero eigenvalues (non-zero curva- 1.3 Laplacian of electron density tures of ρ(rc) at the CPs) and signature is the algebraic sum of the signs of eigenvalues (signs of curvatures of In the topological analysis of ED, the Laplacian ∇2ρ ρ(rc) at the CPs) There are four possible values for CPs ( ) plays a very important role in the character- of rank three. They are (3, −3), (3, −1), (3, +1) and ization of chemical bonding. In fact, the ∇2ρ pro- (3, +3). In the case of (3, −3), all curvatures are neg- vides physical basis for the celebrated electron pair 33,34 ative and ρ is a local maximum at rc.For(3,−1), the model of Lewis. It can be combined with other two curvatures are negative and ρ is a maximum at rc important concepts in electronic structure theory of is in the plane defined by their corresponding axes. Fur- molecules. For example, ∇2ρ along with the electro- ther, ρ is a minimum at CP along the third axis, which static Hellmann-Feynman theorem facilitates the char- is perpendicular to this plane. The (3, +1) CP has two acterization of binding or non-binding with respect to 2 2 positive curvatures and ρ is a minimum at CP in the a given interaction in a molecule. The ∇ ρ(rc)<0 plane defined by their corresponding axes. Further, ρ is indicates the concentration of charge towards interac- a maximum at CP along the third axis, which is per- tion line. The concentration of charge leads to contrac- pendicular to this plane. The (3, +3) CP represents all tion of ρ(rc) perpendicular to the interaction line and curvatures are positive and ρ is a local minimum at rc. lowers the potential energy. The magnitude of lowering It is evident from the topological distribution of of the potential energy is greater than the kinetic energy ED of benzene that (3, −3) CP occurs at the nuclear from the same region thereby creating attractive force 2 positions. In the QTAIM parlance, the nuclei act as and bound shared interaction. The ∇ ρ(rc)>0 implies the attractors of the gradient of ED distribution of that the interaction is dominated by the contraction of molecules. The basin of attractor is a region of three- ρ(rc) towards each nucleus. The parallel gradient and dimensional space, which extends throughout all the the curvature of ρ(rc) are large. In this case net forces space. An atom is defined as the union of attractor and of repulsion act on the nuclei. 1530 P Shyam Vinod Kumar et al.

It is evident from literature that there are other impor- This concept can be used to gain insight into changes tant relationships between energetic topological param- in the atomic properties upon bonding with other sys- 2 eters and the ∇ ρ(rc) at CPs. One of the important tems. One of the important properties of atoms in relationships is the local form of virial theorem:2 molecule is the net charge on an atom.2  1 2 ∇ ρ(rc) = 2G(rc) + V(rc) (5) N() = ρ(r)dτ (9) 4  H(rc) = V(rc) + G(rc) (6) q() = (Z − N ()) e (10) where G(rc), V(rc),andH(rc) are the kinetic energy, potential energy, and the total electron energy densi- where, N(), q() and Z are average number of ties respectively. G(rc) is a positive quantity and V(rc) electrons, net charge on an atom and atomic number is a negative quantity. The balance between the kinetic respectively. electron energy density G(rc) and the potential elec- If A equals the radial distance of an electron from the tron energy density V(rc) reveals the nature of the nucleus, it yields corresponding average over the charge V( interaction. If |V(rc)| > 2G(rc), then the interaction density. Using this definition, the atomic volume ) is covalent in nature. If |V(rc)| is one time more than can be calculated as a measure of region enclosed by 2 the G(rc) then ∇ ρ(rc) is positive and H(rc) is neg- the intersection of its interatomic surfaces and an enve- 2 ative. In this situation, both ∇ ρ(rc) and H(rc) have lope of charge density of some value. The first moment been used to characterize bonding interaction. Simi- M() of an atom’s charge distribution can be obtained G( ) 2 − rc from the following equation: larly, the ratio of V( ) has also been employed to clas- rc  sify the bonding interaction.35 If this ratio is greater M() =− rρ(r)dτ (11) than 1 then the nature of the interaction is purely non-  covalent. Different criteria for describing the nature where r is vector from the nucleus. of bonding using the topological parameters such as ρ( ), ∇2ρ( ) V( ) G( ) H( ) The other important atomic property is quadrupole rc rc , rc , rc ,and rc have been 2 moment, Qzz().Itisdefinedas, summarized in previous reports.3,36 

2 2 Qzz() =− 3z − r ρ(r)dτ (12) 1.4 Integrated atomic properties  The changes in these atomic properties after bonding The underlying fundamental concept in QTAIM is the provide finer details on the nature of bonding. Further- topology of ED. The topology of ED determines the more, it is possible to transfer these atomic or group natural partitioning of the molecular space into disjoint properties to other systems to develop appropriate rela- regions, which are identified as atoms in molecules.37 tionship. Popelier made very detailed analysis of these The atoms in molecules are bounded by surface of zero- properties for hydrogen bonded complexes38,39 and flux in the gradient vector field of ED as given in the added further criteria involving these integrated prop- following equation,2 erties of atoms in molecules. They are, (i) an increase ∇ρ(r).n(r) = 0(7)in net charge, (ii) an energetic destabilization, (iii) a decrease in dipolar polarization, and (iv) a decrease where, ∇ρ(r) is the gradient of the electron density in atomic volume for the proper characterization of and n(r) is a unit vector normal to the surface. Thus, hydrogen bonding interaction in addition to the correct with the help of atoms in molecule partitioning con- topological pattern. cept, it is possible to define atomic properties of atoms in molecule. The atomic average of an observable Aˆ is defined2 as 2. Characterization of Chemical Bonds       N   A() ≡ Aˆ = dτ dτ ψ ∗Aψˆ + (Aψ)ˆ ∗ψ Bader and Essen have reported that the hallmark of   2 “shared (covalent)” interactions is the high value of (8) the ED at BCP of order >10−1 a.u. and negative value  2 12a 2 where,  is the sub-space, dτ and dτ are the volume of ∇ ρ(rc). Thenegativevalueof∇ ρ(rc) indicates elements and N is the total number of electrons. The that there is a concentration of electronic charge at the most important outcome of the above definition is BCP, which indicates the covalent nature of the bond. the average value of any observable of the molecules, The molecular graphs of water, ethane, cyclopropane which can be calculated from the ED. and cubane are depicted in Figure 2 along with the Atoms in Molecules and Chemical Bonding 1531

2 AB values of ρ(rc) and ∇ ρ(rc). These values are in agree- using the dimer basis set, EB is the total energy of ment with the standard values stipulated for covalent the monomer B calculated using the dimer basis set and interaction between two atoms. Both high value of Eint(AB) is the IE of the complex AB. The IEs were 2 ED and the negative value of ∇ ρ(rc) characterize the corrected for Basis Set Superposition Error (BSSE) covalent bonds in these chosen model systems. employing counterpoise method. All electronic struc- ture calculations were performed using GAUSSIAN 09 43 2.1 Characterization of weak interaction package. The wave function generated from these cal- culations were provided as the input for the QTAIM The theory of QTAIM provides an elegant approach analysis. The QTAIM analysis was carried out employ- to unravel the intermolecular interactions with the help ing AIM200044 and Multiwfn45 software packages. 2 24,27,40 of ρ(rc) and ∇ ρ(rc). Various classes of interac- To demonstrate the power of QTAIM in describing tions include weak van der Waals and hydrogen bond- weak van der Waals interaction, benzene dimer has ing interactions. For these interactions, ρ(rc) is quite been chosen as one of the examples. Several electronic small (∼10−2 a.u. or less for H-bonded complexes and structure calculations and experimental studies46,47 have −3 2 10 a.u. for van der Waals complexes) and ∇ ρ(rc) been performed on the benzene dimer to model π-π is positive. To illustrate the power of the QTAIM, we interaction in biological systems.48 50 Despite several have considered a few examples for weak van der intensive efforts on this system, it was difficult to obtain Waals, hydrogen bonded and ionic interactions. The the definitive structure of the benzene dimer. The role geometries of all these inter-molecular complexes were of C-H...π and π...π interactions in the stabilization optimized using MP2/6-311++G** level of theory.41 of T-shaped and parallel-displaced (PD) structures has The interaction energies (IEs) of these complexes were been addressed in previous investigations.21 It is now calculated using supermolecule approach at MP4/6- well established by theoretical and experimental inves- 311++G** level42 using the following equation. tigations that benzene dimer exists in the T-shaped and 51,52 AB AB AB PD structures. The role of electron correlation in Eint(AB) = E − E − E (13) AB A B the stabilization of these complexes has been addressed. AB where, EAB is the total energy of the complex AB, Classically, the electrostatic quadrupole-quadrupole inter- AB EA is the total energy of the monomer A calculated action stabilizes the T-shaped structure when compared

Figure 2. Molecular graphs of water, ethane, cyclopropane and cubane systems with their electron densities and their Laplacian values in a.u. The small red, yellow and green dots represent the bond critical points, ring critical points and the cage critical points, respectively. 1532 P Shyam Vinod Kumar et al. to the PD structure. Hohenstein and Sherill have shown quadrupole-quadrupole interaction.54 The optimized that the T-shaped and PD-structures are nearly isoener- geometry of acetylene dimer along with molecular getic.53 The optimized geometries of benzene dimer graph is depicted in Figure 4. The calculated IE for along with the molecular graphs are given in Figure 3. acetylene dimer is −0.93 kcal/mol. The calculated ++ 2 The calculated IEs at MP4/6-311 G** level of the- ρ(rc) and ∇ ρ(rc) are 0.0078 and 0.0062 a.u., respec- ory reveal that T-shaped structure (−1.60 kcal/mol) is tively. These values are akin to the range proposed in the marginally more stable than the PD-structure (−1.20 criteria for the characterization of weak interactions.39 ...π kcal/mol). The presence of C-H interaction in T- According to the criteria, the ρ(rc) values at the BCP 39 2 shaped structure is evident from the molecular graph. range from 0.002–0.034 a.u. Similarly, the ∇ ρ(rc) The presence of RCP and CCP can be seen in the molec- values at the BCPs vary from 0.024–0.139 a.u.39 ular graphs. The existence of weak interaction between The hydrogen bonding interaction in water clus- the benzene units is evident from the values of ED and ters has been the subject of numerous experimental 2 ∇ ρ(rc). and theoretical investigations due to their implications Another example considered here in the weak bonding in chemistry and biology.4,55,56 The prototype model category is the acetylene dimer. Similar to benzene, for the description of hydrogen bonding interaction is it exhibits T-shaped structure due to the presence of water dimer. Therefore, water dimer has been chosen as

Figure 3. Optimized geometries and molecular graphs of benzene dimers (T-Shaped & Parallel Displaced) at MP2/6- 311++G** level of theory. The electron density and Laplacian values are in a.u. The small red, yellow and green dots represent the bond critical points, ring critical points and the cage critical points, respectively. The IEs were calculated at MP4/6-311++G** level of theory.

Figure 4. Optimized geometry and molecular graph of acetylene dimer at MP2/6-311++G** level of theory. The electron density and its Laplacian val- ues are in a.u. The small red dots represent the bond critical points. The IE is calculated at the MP4/6-311++G** level of theory. Atoms in Molecules and Chemical Bonding 1533 the model system to describe hydrogen-bonding inter- (HCOOH)2 and (CH3COOH)2 dimers are −13.39 and action with help of the QTAIM. The optimized geometry −14.40 kcal/mol, respectively. The optimized geome- of water dimer along with the ED contour map, relief tries and molecular graphs of these dimers are shown map and molecular graph are given in Figure 5. in Figure 6. It is evident from the molecular graph It is evident from the ED topological parameters that that there is a CP between hydrogen (oxygen) of one ρ(rc) at the HBCP is higher than the values obtained HCOOH (CH3COOH) and oxygen (hydrogen) of other for the benzene and acetylene dimers. It can be noticed HCOOH (CH3COOH). Thus, the presence of donor that the ρ(rc) at the BCP increases when we move acceptor interaction in hydrogen-bonded complexes can from weak van der Waals interaction to moderately be clearly seen from the molecular graphs and associ- strong hydrogen bonding interaction. In this category, ated topological parameters. we have also undertaken QTAIM analysis on formic It is evident from the previous reports that QTAIM acid and acetic acid dimers. The calculated IEs of is highly useful to characterize the ionic molecular

Figure 5. Optimized geometry, molecular graph and electron density contour of water dimer at MP2/6-311++G** level of theory. The electron density and its Laplacian values are in a.u. The small red dots represent the bond critical points. The IE is calculated at the MP4/6-311++G** level of theory.

Figure 6. Optimized geometries, molecular graphs and relief maps of formic acid and acetic acid dimmers, respectively at MP2/6-311++G** level of theory. The electron densities and their Laplacian values are in a.u. The small red and yellow dots in molecular graphs represent the bond critical points and ring critical points, respectively. The IEs were calculated at MP4/6-311++G** level of theory. 1534 P Shyam Vinod Kumar et al. clusters.3,30 The strength of ionic hydrogen bonds ranges bonds involves partial proton transfer from donor to from ∼5.0 to 35.0 kcal/mol. These interactions are the acceptor. When a proton interacts with a single implicated in ionic crystals and clusters, ion solvation, water molecule, it forms a strong covalent bond with the + electrolytes and acid-base chemistry. The importance of oxygen to form the hydronium ion (H3O ), known this interaction in proton solvation, surface phenome- as Eigen cation. It is a key species in the transfer of non, self-assembly process in supramolecular chem- proton between molecules in the aqueous acid base istry and bio-molecular structure and function has also conditions. Thus, the QTAIM theory has been applied been recognized. The importance of charge-assisted to characterize the hydrogen bonding interaction in the hydrogen bonds in these systems has been addressed in step-wise solvation of hydronium ion.55 The opti- 55 58 + previous studies. The formation of ionic hydrogen mized geometries and molecular graphs of H3O ...Wn

Figure 7. Optimized geometries and molecular graphs of protonated water clusters at MP2/6-311++G** level of theory. The electron density and Laplacian values are in a.u. The small red dots represent the bond critical points. The IEs were calculated at MP4/6-311++G** level of theory. Atoms in Molecules and Chemical Bonding 1535

Table 1. The interaction energies (IE) of protonated water presented. It is evident from the results on model sys- clusters along with their H-bond distances calculated at tems that QTAIM is an immensely useful tool to char- ++ MP4/6-311 G** level of theory. acterize the nature of bonding. In addition to the correct Cluster Number of H-bonds IE IE (kcal/mol)/ topological pattern of ED, the integrated properties of (distances in Å) (kcal/mol) H-bond atoms in molecules provide further finer details on the bonding. + H3O ...W1 1 (1.2) −49.2 −49.2 + H3O ...W2 2 (1.5) −57.3 −28.7 + Acknowledgements H3O ...W3 3 (1.5) −74.1 −24.7 + − . − . H3O ...W6 6 (1.5-1.7) 109 1 18 2 Authors would like to thank the Council of Scientific and Industrial Research (CSIR), New Delhi and Depart- (n=1,2,3,6) clusters are shown in Figure 7. The cal- ment of Science and Technology, India for the funding. 2 culated ED and its ∇ ρ(rc) at HBCPs for various clusters are collected in Table 1. References + The value of ρ(rc) for H3O ...W1 is 0.16 a.u. The addition of the second water molecule results in a sud- 1. Bohórquez H J, Boyd R J and Matta C F 2011 J. Phys. den decrease in the ED values at the HBCP. Marginal Chem. A 115 12991 changes in ρ(r ) can be noticed from the values shown 2. (a) Bader R F W 1990 In Atoms in Molecules: A Quan- c tum Theory (Oxford: Clarendon Press); (b) Bader R F W in Figure 7 for the addition of the third water molecule 1985 Acc. Chem. Res.18 9; (c) Bader R F W 1991 Chem. to the hydronium ion. To predict the strength of the Rev. 91 893 H-bond formed in the second solvation shell, three 3. (a) Grabowski S J 2011 Chem. Rev. 111 2597; (b) more water molecules are added to form the structure Grabowski S J 2012 J. Phys. Chem. A 116 1838 H O+ ...W6 as depicted in Figure 7. The calculated 4. Parthasarthi R, Subramanian V and Sathyamurthy N 3 2005 J. Phys. Chem. A 109 843 ρ(r ) values for the second solvation shell H-bonds are c 5. (a) Arputharaj D S, Hathwar V R, Row T N G and found to be of the order of 0.03 a.u., which is approx- Kumaradhas P 2012 Cryst. Growth Des. 12 4357; (b) imately half of what has been observed for the first Hathwar V R, Paul A V, Natarajan S and Row T N G 2 hydration shell. It is interesting to note that the ∇ ρ(rc) 2011 J. Phys. Chem. A 115 12818; (c) Pavan M S, Pal R, + Nagarajan K and Row T N G 2014 Cryst. Growth Des. at the HBCP is negative for the H3O ...W1 cluster, indicating the covalent character of the bond. 14 5477 ∇2ρ( ) 6. Hirano Y, Takeda K and Miki K 2016 Nature 534 281 For other H-bonds, the rc is positive, implying 7. (a) Parthasarathi R, Amutha R, Subramanian V, Nair the presence of conational hydrogen bonding inter- B U and Ramasami T 2004 J. Phys. Chem. A 108 3817; action. The strength of the first H-bond formed in (b) Parthasarathi R and Subramanian V 2005 Struct. + H3O ...W1 is ∼50.0 kcal/mol, which is in accor- Chem. 16 243 dance with its covalent character. The SE per H-bond 8. Szabo A and Ostlund N S 1989 In Modern Quan- + tum Chemistry: Introduction to Advanced Electronic in H3O ...W2 is ∼29.0 kcal/mol, in agreement with ρ( ) Structure Theory (New York: Dover Publications) the decrease in the corresponding rc values. For the 9. Popelier P 2000 In Atoms in Molecules: An Introduction + ... completed first solvation shell structure H3O W3, (New York: Prentice Hall) SE per H-bond is ∼25.0 kcal/mol, and this is in accor- 10. (a) Brown E C, Bader R F W and Werstiuk N H 2009 dance with the corresponding decrease in the ρ(rc) J. Phys. Chem. A 113 3254; (b) Bone R G A and Bader values. SE per H-bond in the second solvation shell R F W 1996 J. Phys. Chem. 100 10892 11. (a) Bader R F W 2009 J. Phys. Chem. A 113 10391; (b) is ∼19.0 kcal/mol, which is reflected in the decrease ρ( ) Bader R F W 2010 J. Phys. Chem. A 114 7431 in the rc value at various HBCPs. These results 12. (a) Bader R F W and Essén H 1984 J. Chem. Phys. 80 2 demonstrate that ρ(rc) and ∇ ρ(rc) values at the HBCP 1943; (b) Bader R F W and Matta C F 2001 Inorg. Chem. provide meaningful information to characterize hydro- 40 5603 gen-bonding interactions in different solvation shells. 13. (a) Bader R F W and MacDougall P J 1985 J. Am. Chem. It can also be seen that ∇2ρ(r ) is a useful parame- Soc. 107 6788; (d) Bader R F W and Fang D-C 2005 J. c Chem. Theory Comput. 1 403 ter to differentiate conventional hydrogen bonding vs. 14. Chaudry U A and Popelier P L A 2004 J. Org. Chem. 69 covalent bonding in molecular clusters. 233 15. Devereux M and Popelier P L A 2007 J. Phys. Chem. A 111 1536 3. Summary 16. Yuan Y, Mills M J L, Popelier P L A and Jensen F 2014 J. Phys. Chem. A 118 7876 In this perspective, the basic theory and applications 17. Griffiths M Z and Popelier P L A 2013 J. Chem. Inf. of quantum theory of atoms in molecules have been Model. 53 1714 1536 P Shyam Vinod Kumar et al.

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